Question

In the given diagram, the points A, B, and C lie on a circle whose center is O. BOC is a diameter of the circle.

The following set of premises is relevant to the investigation of the given geometric diagram:

I. The radii of a circle are of equal length.
II. Angles opposite to equal sides of a triangle are equal.
III. The sum of the angles of a triangle is 180°.

With reference to the diagram, which conclusion can be drawn correctly from the premises?

Answer 1

m∠BAC = 90°

Option B

Explanation:

From premise 1 we get AO = AC and AO = OB
From premise 1 and 2 we get
m∠OAC = m∠OCA = x say and
m∠OAB = m∠OBA = y say

But m∠OCA = m∠ACB =x
m∠OBC = m∠ABC = y

From all three premises we get in triangle ABC:

m∠BAC + m∠ABC+ m∠ACB = 180

Knowing
m∠BAC = m∠OAC + m∠OAB = x + y

m∠ACB = x

m∠ABC = y

So we get

(x + y) + x + y = 180

2x + 2y = 180
x + y = 180/2 = 90°

Since x + y = m∠BAC,

m∠BAC = 90°